Q. 20 - Calculus

Question

Each of the definite integrals in Exercises 19–24 represents the volume of a solid of revolution obtained by rotating a region around either the x- or y-axis. Find this region. 

$\mathrm{\pi }{\int }_0^{\mathrm{\pi }}{\sin }^2\mathrm{xdx}$

Step-by-Step Solution

Answer

The required region is shown as, 

Step 1. Given Information

The given integral is $\mathrm{\pi }{\int }_0^{\mathrm{\pi }}{\sin }^2\mathrm{xdx}$

Step 2. Explanation

The given integral is of the form $\mathrm{\pi }{\int }_a^b(f{\left(x\right))}^2\mathrm{dx}$ which represents the volume of a solid formed by rotating the region bound by f(x) and x-axis in the interval $[a,b]$ around x-axis.

On comparing the given region with the above region, we have,

$f\left(x\right)=\sin x \mathrm{in} \mathrm{the} \mathrm{interval} [0,\mathrm{\pi }]$

The given function represents the sine curve.

Plot the graph of the function and identify the region between the intervals.